COMP3609 Algorithms and Complexity (Advanced)
Edmund Tse 2008
Since xi* + xj* 1, either xi* or xj* (i, j) covered.
Pf. [S has desired cost]
Let S* be optimal vertex cover. Then
o
Theorem. 2-approximation for weighted vertex cover
o
o
Theorem. [Dinfur-Sa

COMP3609 Algorithms and Complexity (Advanced)
Edmund Tse 2008
Pricing method. Set prices and find vertex cover simultaneously
Pricing method analysis
o Theorem. Pricing method is a 2-approximation
o Pf.
Algorithm terminates since at least one new node be

COMP3609 Algorithms and Complexity (Advanced)
o
o
o
Edmund Tse 2008
Observation. If at most m jobs, then list-scheduling is optimal
Pf. Each job put on its own machine
Lemma. If there are more than m jobs, L* 2 tm+1
Consider first m + 1 jobs t1, , tm+1

COMP3609 Algorithms and Complexity (Advanced)
Edmund Tse 2008
Wavelength-division-multiplexing (WDM). Allows m communication streams (arcs) to share a portion
of a fibre optic cable, provided they are transmitted using different wavelengths
o Ring topolog

COMP3609 Algorithms and Complexity (Advanced)
Edmund Tse 2008
Find median of the medians
Partition around the median of the medians O(n)
Sort each group O(n)
If not done, continue recursively on one side of the partition
+6
The master theorem
o
Let ( )

COMP3609 Algorithms and Complexity (Advanced)
Edmund Tse 2008
The Median Algorithm and Analysis of Quicksort
Median: the
The peak in a unimodal array
o Given an array A with n entries, with each entry holding a distinct number
o The sequence A[1], A[2], ,

COMP3609 Algorithms and Complexity (Advanced)
Edmund Tse 2008
Chapter 11 Approximation Algorithms
Theory says it is unlikely to find a poly-time algorithm for an NP-hard problem.
Must sacrifice one of three desired features: optimality, poly-time or abili